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:''See also Wigner distribution (disambiguation).'' The Wigner quasiprobability distribution (also called the Wigner function or the Wigner–Ville distribution after Eugene Wigner and Jean-André Ville) is a quasiprobability distribution. It was introduced〔E.P. Wigner, "On the quantum correction for thermodynamic equilibrium", ''Phys. Rev.'' 40 (June 1932) 749–759. 〕 by Eugene Wigner in 1932 to study quantum corrections to classical statistical mechanics. The goal was to link the wavefunction that appears in Schrödinger's equation to a probability distribution in phase space. It is a generating function for all spatial autocorrelation functions of a given quantum-mechanical wavefunction . Thus, it maps〔H.J. Groenewold, "On the Principles of elementary quantum mechanics",''Physica'',12 (1946) 405–460. 〕 on the quantum density matrix in the map between real phase-space functions and Hermitian operators introduced by Hermann Weyl in 1927,〔H. Weyl, ''Z. Phys.'' 46, 1 (1927). ; H. Weyl, ''Gruppentheorie und Quantenmechanik'' (Leipzig: Hirzel) (1928); H. Weyl, ''The Theory of Groups and Quantum Mechanics'' (Dover, New York, 1931).〕 in a context related to representation theory in mathematics (cf. Weyl quantization in physics). In effect, it is the Wigner–Weyl transform of the density matrix, so the realization of that operator in phase space. It was later rederived by Jean Ville in 1948 as a quadratic (in signal) representation of the local time-frequency energy of a signal,〔J. Ville, "Théorie et Applications de la Notion de Signal Analytique", ''Câbles et Transmission'', 2, 61–74 (1948).〕 effectively a spectrogram. In 1949, José Enrique Moyal, who had derived it independently, recognized it as the quantum moment-generating functional,〔 J.E. Moyal, "Quantum mechanics as a statistical theory", ''Proceedings of the Cambridge Philosophical Society'', 45, 99–124 (1949). 〕 and thus as the basis of an elegant encoding of all quantum expectation values, and hence quantum mechanics, in phase space (cf. phase space formulation). It has applications in statistical mechanics, quantum chemistry, quantum optics, classical optics and signal analysis in diverse fields such as electrical engineering, seismology, time–frequency analysis for music signals, spectrograms in biology and speech processing, and engine design. == Relation to classical mechanics == A classical particle has a definite position and momentum, and hence it is represented by a point in phase space. Given a collection (ensemble) of particles, the probability of finding a particle at a certain position in phase space is specified by a probability distribution, the Liouville density. This strict interpretation fails for a quantum particle, due to the uncertainty principle. Instead, the above quasiprobability Wigner distribution plays an analogous role, but does not satisfy all the properties of a conventional probability distribution; and, conversely, satisfies boundedness properties unavailable to classical distributions. For instance, the Wigner distribution can and normally does go negative for states which have no classical model—and is a convenient indicator of quantum mechanical interference. Smoothing the Wigner distribution through a filter of size larger than (e.g., convolving with a phase-space Gaussian, a Weierstrass transform, to yield the Husimi representation, below), results in a positive-semidefinite function, i.e., it may be thought to have been coarsened to a semi-classical one.〔Specifically, since this convolution is invertible, in fact, no information has been sacrificed, and the full quantum entropy has not increased, yet. However, if this resulting Husimi distribution is then used as a plain measure in a phase-space integral evaluation of expectation values ''without the requisite star product of the Husimi representation'', then, at that stage, quantum information ''has been forfeited'' and the distribution ''is a semi-classical one'', effectively. That is, depending on its usage in evaluating expectation values, the very same distribution may serve as a quantum or a classical distribution function.〕 Regions of such negative value are provable (by convolving them with a small Gaussian) to be "small": they cannot extend to compact regions larger than a few , and hence disappear in the classical limit. They are shielded by the uncertainty principle, which does not allow precise location within phase-space regions smaller than , and thus renders such "negative probabilities" less paradoxical. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Wigner quasiprobability distribution」の詳細全文を読む スポンサード リンク
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